In Europe and the Mediterranean, really strong, clear red and blue dyes were hard to create, too. Turkey red is a nice, strong red, but the dyeing process was a closely-guarded secret for a long time. Kermez or crimson (a red pigment made from scale insects, was known, but it’s labor-intensive and expensive to produce (like Tyrian purple!). In Europe, there are plants that will produce red dye (like madder), but it’s not a bright red, and you need a bright red to get a vibrant purple.
Indigo and woad will produce a nice blue that you can use to get green and purple. However, you still have to deal with red – earthy or dull red will produce a muted purple when overdyed with indigo or woad.
After Europeans began colonizing the new world, they had a few more options. Cochineal is a bright, pinkish-red dye made from crushed cactus beetles. It can be overdyed with indigo to produce a good purple. Logwood can produce a purplish color that can be enhanced with cochineal or indigo.
Synthetic dyes weren’t available until the 19th century. Mauve – a purple color – was one of the first commercially successful synthetic dyes, and it was _wildly_ popular.
Edit: Forgot about kermez
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